Lecture on Drawing Cubic Graphs

Introduction to Cubic Graphs

• General equation of a cubic: ( y = ax^3 + bx^2 + cx + d )
  • ( a \neq 0 ) ensures the presence of the ( x^3 ) term.
• Examples:
  • ( y = x^3 )
  • ( y = -x^3 )
  • ( y = 2x^3 + ax^2 - 7x + 3 )
  • ( y = x^3 + 4x - 2 )
• A cubic function may or may not include ( x^2 ), ( x ), or constant terms.

Graph of ( y = x^3 )

• Shape: Comes up, flattens at the origin, and curves up steeply.
• Behavior:
  • Right of y-axis:
    • 0: Passes through origin.
    • 1: Point (1,1)
    • 2: Point (2,8)
    • 3: Point (3,27)
    • 4: Point (4,64)
  • Left of y-axis:
    • -1: Point (-1,-1)
    • -2: Point (-2,-8)
    • -3: Point (-3,-27)
• Observation: Steep curve going up to the right or down to the left.

Graph of ( y = -x^3 )

• Shape: Comes down, flattens, and curves sharply downwards.
• Behavior:
  • Negative reflection of ( y = x^3 ).
• Points:
  • 1: Point (1,-1)
  • 2: Point (2,-8)
  • 3: Point (3,-27)

Complex Cubics with Extra Terms

Example 1: ( y = x^3 + 2x^2 - 3 )

• Table of Values:
  • 2: (2,13)
  • 1: (1,0)
  • 0: (0,-3)
  • -1: (-1,-2)
  • -2: (-2,-3)
  • -3: (-3,-12)
• Shape: Comes up, flattens, goes down, and then up.

Example 2: ( y = -x^3 + 3x^2 + x - 1 )

• Table of Values:
  • 4: (4,-13)
  • 3: (3,2)
  • 2: (2,5)
  • 1: (1,2)
  • 0: (0,-1)
  • -1: (-1,2)
  • -2: (-2,17)
  • -3: (-3,50)
• Shape: Comes down, curves up, and then down again.

Common Shapes of Cubic Graphs

• Simple ( x^3 ) or ( -x^3 ): Basic upward or downward steep curves.
• Complex Forms:
  • Positive ( x^3 ): Up, down, up again (has extra terms).
  • Negative ( x^3 ): Down, up, down again (has extra terms).

Conclusion

• Understanding these forms and how to calculate key points is crucial for drawing accurate cubic graphs.